# Greedy Monochromatic Island Partitions

CoRR（2024）

Abstract

Constructing partitions of colored points is a well-studied problem in
discrete and computational geometry. We study the problem of creating a
minimum-cardinality partition into monochromatic islands. Our input is a set
S of n points in the plane where each point has one of k ≥ 2 colors. A
set of points is monochromatic if it contains points of only one color. An
island I is a subset of S such that 𝒞ℋ(I) ∩ S = I, where
𝒞ℋ(I) denotes the convex hull of I. We identify an island with
its convex hull; therefore, a partition into islands has the additional
requirement that the convex hulls of the islands are pairwise-disjoint. We
present three greedy algorithms for constructing island partitions and analyze
their approximation ratios.

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